Mathematische Annalen

, Volume 370, Issue 1–2, pp 819–839 | Cite as

An equivariant parametric Oka principle for bundles of homogeneous spaces

  • Frank Kutzschebauch
  • Finnur Lárusson
  • Gerald W. Schwarz


We prove a parametric Oka principle for equivariant sections of a holomorphic fibre bundle E with a structure group bundle \({{\mathscr {G}}}\) on a reduced Stein space X, such that the fibre of E is a homogeneous space of the fibre of \({{\mathscr {G}}}\), with the complexification \(K^{{\mathbb {C}}}\) of a compact real Lie group K acting on X, \({{\mathscr {G}}}\), and E. Our main result is that the inclusion of the space of \(K^{{\mathbb {C}}} \hbox {-equivariant}\) holomorphic sections of E over X into the space of \(K\hbox {-equivariant}\) continuous sections is a weak homotopy equivalence. The result has a wide scope; we describe several diverse special cases. We use the result to strengthen Heinzner and Kutzschebauch’s classification of equivariant principal bundles, and to strengthen an Oka principle for equivariant isomorphisms proved by us in a previous paper.

Mathematics Subject Classification

Primary 32M05 Secondary 14L24 14L30 32E10 32Q28 



We thank Michael Murray for help with the theory of generalised principal bundles.


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Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  • Frank Kutzschebauch
    • 1
  • Finnur Lárusson
    • 2
  • Gerald W. Schwarz
    • 3
  1. 1.Institute of MathematicsUniversity of BernBernSwitzerland
  2. 2.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia
  3. 3.Department of MathematicsBrandeis UniversityWalthamUSA

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